We develop a new Bayesian Markov Chain Monte Carlo algorithm for Euler-discretized Feller square-root stochastic volatility models and demonstrate the performance of our algorithm through simulations and empirical analyses. Specifically, our algorithm use the Laplace approximation of the posterior density of conditional variance, which is the probability kernel of the generalized inverse gaussian distribution, derived from the joint density of return and conditional variance so that it can be easily applied to the extended stochastic volatility models with such as fat-tailed distributions or Levy jump processes. In addition, we conduct the simulation experiment investigating and comparing the size and power of the parametric specification tests checking certain finite-dimensional moment conditions without correction for parameter estimation uncertainty with that of the nonparametric Hong and Li (2005)’s omnibus test which is not affected by parameter estimation uncertainty. The parametric and nonparametric tests are based on the probability integral transform of the prediction densities of returns obtained using auxiliary particle filter algorithms. Our experiment result shows that the classical parametric specification test may have no worse size distortion and better power than Hong and Li (2005)’s test.